math 0030 lecture notes section 2.1 the addition property ... · pdf filemath 0030 –...
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Introduction
Most, but not all, salaries and prices have soared over the decades. To make it easier to
compare, the figure below converts historical prices into today’s dollars, with adjustments based
on consumer’s price index.
The data for the average annual salary can be described by the mathematical model
262 12000S n
where S is the average U.S. annual salary, in dollars, n years after 1901.
If trends shown in the formula continue, when will the average annual salary be $50,000? Round
to the nearest year.
MATH 0030 – Lecture Notes
Section 2.1 The Addition Property of Equality
Section 2.2 The Multiplication Property of Equality
Solving an equation is the process of finding the ______________ or ___________________ that
make the equation statement _______________.
Circle the linear equations in one variable.
3 7 9x 23 7 9x 15
45x
15 45x 6.8x 6.8x
Example: Solve the equations.
a. 5 12x b. 2.8 5.09z c. 1 3
2 4x
Example: Solve by combining like terms before using the addition property.
8 7 7 10 6 4y y
Example: Solve by isolating the variable to one side of the equation.
3 6 2 5x x
Linear equation in one variable x is an equation that can be written in the form
ax b c
where a, b, and c are real numbers and a is not equal to 0
The Addition Property of Equality
The same real number (or algebraic expression) may be added to both sides of an equation
without changing the equation’s solution. This can be expressed symbolically as follows:
If a = b, the a + c = b + c
Example: Solve the equations.
a. 11 44y b. 15.5 5z c. 2
163
y d. 3x
Example: Solve using both the addition and multiplication properties.
a. 4 3 27x b. 4 15 25y c. 2 15 4 21x x
The Multiplication Property of Equality
The same nonzero real number (or algebraic expression) may multiply both sides of an
equation without changing the solution. This can be expressed symbolically as follows:
If a = b and 0c , then ac = bc.
Introduction:
In Massachussetts, speeding fines are determined by the formula
10( 65) 50F x
where F is the cost in dollars, of the fine if a person is caught driving x miles per hour.
Use this formula to solve, if a fine comes to $400, how fast was the person driving?
Example: Solve and check.
a. 7 25 3 16 2 3x x x b. 8 2( 6)x x c. 4(2 1) 29 3(2 5)x x
MATH 0030 - Lecture Notes
Section 2.3 Solving Linear Equations
Steps to Solving a Linear Equation.
1. Simplify the algebraic expression on each side.
2. Collect all the variable terms on one side and all the constant terms on the other side.
3. Isolate the variable and solve.
4. Check the proposed solution in the original equation.
Example: Solve by clearing fractions.
a. 3 39
2 5 5
x x b.
2 5
4 3 6
x x
0.3 10 0.37 100 0.408 1000
Example: Solve by clearing decimals.
a. 0.3 6 0.37 1.1x x b. 0.48 3 0.2 6x x
Example: Solve.
a. 3 7 3 1x x b. 3 1 9 8 6 5x x x
Clearing Equations with Fractions: We begin by multiplying _____________ sides of the
equation by the _________________________ of all fractions in the equation. The LCD is the
______________________ number that all the denominators will divide into. Multiplying
________________ term on both sides of the equation by the least common denominator will
eliminate the fractions in the equation.
Clearing Equations with Decimals: It is not necessary with a calculator, but multiplying a
decimal number by a power of _______________, has the effect of moving the decimal place to
the right.
Recognizing Inconsistent (No Solution) and Identities (All real Numbers)
If you attempt to solve an equation with no solution or one that is true for every real number,
you will eliminate the variable.
An inconsistent equation with NO SOLUTION results in a FALSE statement, such as 2 5
An identity that is true for ALL REAL NUMBERS results in a TRUE statement, such as 4 4
Solving for a Variable means rewriting the formula so that the variable is ____________________
on one side of the equation. It does not mean obtaining a numerical value for that variable.
Example: Solve the formulas for the indicated variables.
a. A lw solve for l b. 2 2P l w solve for l
c. PrS P t solve for r d. Ax By C solve for y
Example: Write an equation then solve.
a. What is 9% of 50?
b. 9 is 60% of what?
c. 18 is what percent of 50?
MATH 0030 - Lecture Notes
Section 2.4 Formulas and Percents
Formula’s involving Percent:
Percent Increase and Decrease
Percents are used for comparing changes, such as increases and decreases in sales, population,
prices, and production. If a quantity changes, its’ percent increase or percent decrease can be
determined
Example: Write an equation then solve.
a. A charity has raised $7500, with a goal of raising $500,000. What percent of the goal has
been raised?
b. Suppose that the state sales tax rate is 9% and you buy a graphing calculator for $96.
How much tax is due? What is the calculator’s total cost?
Percent Increase: The increase is what percent of the original amount?
Where A, represents the increase, P represents the percent and B represents the original
amount.
Percent Decrease: The decrease is what percent of the original amount?
Where A, represents the decrease, P represents the percent and B represents the original
amount.
Introduction
The bar graph show the ten most popular college majors with median, or middlemost, starting
salaries for recent college graduates.
The median starting salary for a nursing major exceeds that of a business major by $10 thousand.
Combined, their median starting salaries are $96 thousand. Determine the median starting
salaries of business and nursing majors with bachelor’s degrees.
Define the variables in words: x = ____________________________
Write and algebraic equation:
Solve:
Sentence Answer: _______________________________________________________________________
________________________________________________________________________________________
MATH 0030 - Lecture Notes
Section 2.5 An Introduction to Problem Solving
Strategy for Solving Word problems
1. Read the problem carefully. Define variables by letting x represent one of the unknown
quantities.
2. Write an equation in x that translates, or models the conditions of the problem.
3. Solve the equation and answer the problem’s question.
4. Check the solution in the original wording of the problem
English Phrase Algebraic
Expression
Addition
The sum of a number and 7
Five more than a number; a number plus 5
A number increased by 6: 6 added to a number
x + 7
x + 5
x + 6
Subtraction
A number minus 4
A number decreased by 5
A number subtracted from 8
The difference between a number and 6
The difference between 6 and a number
Seven less than a number
Seven minus a number
Nine fewer than a number
x – 4
x – 5
8 – x
x – 6
6 – x
x – 7
7 – x
x – 9
Multiplication
Five times a number
The product of 3 and a number
Two-thirds of a number (used with fractions)
Seventy-five percent of a number(used with decimals)
Thirteen multiplied by a number
A number multiplied by 13
Twice a number
5x
3x
2
3x
0.75x
13x
13x
2x
Division
A number divided by 3
The quotient of 7 and a number
The quotient of a number and 7
The reciprocal of a number
3
x
7
x
7
x
1
x
More than one operation
The sum of twice a number and 7
Twice the sum of a number and 7
Three times the sum of 1 and twice a number
Nine subtracted from 8 times a number
Twenty-five percent of the sum of 3 times a number and 14
Seven times a number, increased by 24
Seven times the sum of a number and 24
2x + 7
2(x+7)
3(1 + 2x)
8x – 9
0.25(3x + 14)
7x + 24
7(x+24)
Example: The bar graph shows that average rent and mortgage payments in the United States
have increased since 1975, even after taking inflation into account.
In 2008, rent payments averaged $824 per month. For the period
shown, monthly rent payments increased by approximately $7
per year. If this trend continues, how many years after 2008 will
rent payments average $929? In which year will this occur?
Define variables: x = __________________________
Equation:
Solve:
Sentence Answer: ________________________________________________________________________
__________________________________________________________________________________________
Example: A rectangular swimming pool is three times as long as it is wide. If the perimeter of
the pool is 320 feet, what are the pool’s dimensions?
Define variables: x = __________________________
Equation:
Solve:
Sentence Answer: ________________________________________________________________________
__________________________________________________________________________________________
Example: After a 40% price reduction, an exercise machine sold for $564. What was the
exercise machine’s price before this reduction?
Define variables: x = __________________________
Equation:
Solve:
Sentence Answer: ________________________________________________________________________
__________________________________________________________________________________________
MATH 0030 - Lecture Notes
Section 2.6 Problem Solving in Geometry
Geometric Formulas for Perimeter and Area
Square Area: 2A s where s is side
Square Perimeter: 4P s where s is side
Rectangle Area: A lw where l is length and w is width
Rectangle Perimeter: 2 2P l w where l is length and w is width
Triangle Area: 1
2A bh where b is base and h is height
Trapezoid Area: 1
2A h a b where b is base and h is height
Geometric Formulas for Circumference and Area of a Circle
Area of Circle: 2A r where r is radius
Circumference of Circle: 2C r where r is radius
Geometric Formulas for Volume
Cube Volume: 3V s where s is side
Rectangular Solid Volume: V lwh where l is length, w is width, and h is
height
Circular Cylinder Volume: 2V r h where r is radius and h is height
Sphere Volume: 34
3V r where r is radius
Cone Volume: 21
3V r h where r is radius and h is height
Circumference of Circle: 2C r where r is radius
Example: A sailboat has a triangular sail with an area of 24 square feet and a base that is 4 feet
long. Find the height of the sail.
Example: The diameter of a circular landing pad for helicopters is 40 feet. Find the area and
circumference of the landing pad. Express answers in terms of . Then round answers to the
nearest square foot and foot respectively.
Example: Which one of the following is the better buy: a large pizza with a 16-inch diameter for
$15.00 or a medium pizza with an 8-inch diameter for $7.50?
Example: A cylinder with a radius of 3 inches and a height of 5 inches has its height doubled.
How many times greater is the volume of the larger cylinder than the volume of the smaller
cylinder?
Example: In a triangle, the measure of the first angle is three times the measure of the second
angle. The measure of the third angle is 20 degrees less than the second angle. What is the
measure of each angle?
Example: The measure of an angle is twice the measure of its complement. What is the angle’s
measure?
Angles of a Triangle: The sum of the measures of the three angles of any triangle is
________________.
Complementary and Supplementary Angles
Complementary Angles are two angles having a sum of ____________________.
Supplementary Angles are two angles having a sum of _____________________.
Algebraic Expressions for Complements and Supplements
Measure of an angle: x
Measure of the angle’s complement: 90-x
Measure of the angle’s supplement: 180-x
Graphing solutions to linear inequalities are shown on a number line by shading all points
representing numbers that are solutions. SQUARE BRACKETS, [ ], indicate endpoints that
____________ solutions. PARENTHESES, ( ), indicate endpoints that ____________________
solutions.
English
Sentence
Inequality Interval
Notation
Set-Builder
Notation
Graph (number lines)
x is more than a x a
,a
x x a
x is at least a x a
,a x x a
x is less than a
x a
,a x x a
x is at most a x a
,a x x a
Example: Solve and graph the solution set on a number line.
a. 8 2 7 4x x b. 6 18x c. 5 3 17y
MATH 0030 - Lecture Notes
Section 2.7 Solving Linear Inequalities
Solving a Linear Inequality
1. Simplify the algebraic expression on each side.
2. Use the addition property of inequality to collect all the variable terms on one side and
all the constant terms on the other side.
3. Use the multiplication property of inequality to isolate the variable and solve. Change
the ___________________ of inequality when multiplying or dividing both sides by a
_________________ number.
4. Express the solution set in interval or set-builder notation, and graph the solution set on
a number line.
Example: Solve and graph the solution set on a number line.
a. 6 3 5 2x x b. 2 3 1 3 2 14x x
Example: Solve
a. 4 2 4 15x x b. 3 1 2 1x x x
Example: You can spend at most $1600 to have a party catered. The caterer charges $95 setup
fee and $35 per person. How many people can you invite while staying within your budget?
Recognizing Inequalities with No Solution or True for all Real Numbers
If you attempt to solve an inequality with no solution or one that is true for every real number,
you will eliminate the variable.
An inequality with no solution results in a _______________ statement, such as 0 >1. The
solution set is , the empty set.
An inequality that is true for every real number results in a ____________________
statement, such as 0<1. The solution set is , or x x is a real number .